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Highways, pathways, and byways in the history of mathematics. Handbook of mathematical formulas and integrals. Unrestricted division by zero as multiplication by the – reciprocal to zero – infinity. Finegrained 3D differential operators hint at the inevitability of their dual reciprocal portrayals. Mathematical gateway to complementary hidden variables in macrophysics. In a sense thus, the process of splitting of spacetime apparently demands a paradigm shift: from the formerly unspoken and thus unchallenged SSR paradigm to a certain MSR paradigm. The conclusion is supported by special relativity. Nevertheless, the separation of spacetime is feasible within the multispatial reality (MSR) framework that allows the ensuing two separated spaces to reside in twin distinct spatial structures each of which is equipped with different orthogonal homogeneous basis, provided the twin spatial structures are equidimensional 3D spaces that are mutually paired to form a hierarchical hyperspatial structure comprising mutually dual reciprocal spaces. It is thus unacceptable because 1/(∂/∂t)=∂t/∂ is a contravariant expression posing as derivative (with respect to some unspecific variable) in the SSR setting and as such cannot be equated to the regular covariant derivative ∂/∂t taken with respect to the independently varying parameter of the elapsing time t. The anticipated separation of the 4D spacetime into 3D length-based space and distinct abstract 1D space of elapsing time, which would vary independently of the usual variables of the length-based space, is not reasonable in the formerly unspoken single space reality (SSR) framework, because the separation would lead to inadmissible identification of multiplicatively inverse/reciprocal derivative taken with respect to elapsing time parameter with the derivative taken with respect to the 1D time variable.
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